Odd/Odd Records for a Relative Approximation Error: Three Quadratic Classifications and an Exact Certificate

We study rational approximation under an odd/odd constraint, ordering positive odd integer pairs (m, k) by the height H = mα + k and measuring their relative error ε = |mα − k|/(mα + k). A strict chronological record improves upon every admissible state of smaller height. For α = (1 + √5)/2, α = 1 + √3, and α = √3, we obtain complete classifications of the strict records as finite unions of explicit unit orbits. The proofs combine exact identities involving a signed quadratic norm and a conjugate factor, height-based norm bounds, exhaustive orbit enumeration, and certified comparisons connecting a finite computation to the infinite sequence. Two distinct equality mechanisms arise. At the golden ratio, a signed correction determines when Lucas pairs improve upon Fibonacci approximations. At √3, an infinite companion family repeats record errors at larger heights, distinguishing weak records from strict records. The results resolve two conjectures from the author’s earlier note on two-mode oscillator record chronologies and clarify the distinction between exact scaling of the conjugate factor and asymptotic scaling of height. An accompanying Python certificate supports reproducibility using exact arithmetic. A uniform classification for general quadratic slopes and congruence classes remains open.